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Prime numbers, prime factors and composite numbers

Prime numbers, prime factors and composite numbers

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Summary

Prime numbers, prime factors and composite numbers

In a nutshell

Prime and composite numbers are special categories of numbers based on how many factors they have. There are 2525 prime numbers and 7474 composite numbers between 11 and 100100.  



Prime numbers and composite numbers

Prime numbers only have two factors, 11 and itself. Composite numbers have more than two factors. 


Example 1

Identify whether the following numbers are prime or composite: 2,10,13,252,10,13,25.


Number

Factors

Prime or Composite

22​​
1,21,2​​

PrimePrime​​

1010​​
1,2,5,101,2,5,10​​

CompositeComposite​​

1313​​
1,131,13​​

PrimePrime​​

2525​​
1,5,251,5,25​​

CompositeComposite​​

​​

Note: 11 is neither a prime or composite number because it only has one factor, 11



Prime factors

Factors of a number which are prime are prime factors. They can be found using a factor tree to breakdown a number into its prime factors which can be multiplied together to give the original number. This can be done by following the procedure below: 


PROCEDURE

11​​

Write down the given number at the top. 

22​​

Find any two numbers which multiply to give the top number. 

33​​

Write down these two factors beneath the original number at the end of a left and right "branch".

44​​

Repeat this process for all the numbers in the tree until each "branch" ends in a prime factor.

55​​

​Circle each prime factor, and write them all multiplied by each other. 


Example 2

What is 3232 as a product of its prime factors?


Write 3232​ on the top, and find two numbers that multiply to give 3232:


32=2×1632=2\times16

​​

Write 22 and 1616 beneath 323222​ is a prime number so circle it.


   32  216{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \end{aligned}}​​


As 1616 is not a prime number, find two numbers that multiply to give 1616 such as: 2×82 \times 8​. Add this to the factor tree and circle 22 since it is prime. ​


   32  216     2 8\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \end{aligned}}

​​

As 88 is not a prime number, find two numbers that multiply to give 88 such as: 2×42 \times 4​. Add this to the factor tree and circle 22 since it is prime.


   32  216     2 8    2  4\quad\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \end{aligned}}​​


​As 44​ is not a prime number, find two numbers that multiply to give 44​ such as 2×22 \times 2​. Add this to the factor tree and circle both 22s since they are prime.

​​

    32  216     2 8    2  4  2  2\quad\quad\quad\space{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \\ &\quad\quad\quad\space\,\,\swarrow\searrow \\ &\quad\quad\quad\textcircled2 \quad\space \,\,\textcircled2\end{aligned}}​​


Only prime factors remain so the tree is complete. 

​​​

Write 3232 as a product of its prime factors.​


Therefore, 2×2×2×2×2=32.\underline{2 \times 2\times 2\times 2\times 2=32.}


Note: 11 is not a prime number so it does not appear in the factor tree.


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